Topics

Introduction to Series - At A Glance:

We can use the idea of partial sums to (finally) describe what it means to add infinitely many numbers together. We can taste our infinite brownie, too. In symbols, we want to know what

means, and just how much chocolaty goodness we are going to get.

Think about what happens when you put laundry detergent into a washing machine. As you add more suds, there's enough to wash the laundry and all goes well. You can also add so much that a detergent monster oozes from the washer, soaping your dry socks and cat Binx.

The same happens with series. As we add numbers, if the sum gets closer and closer to some particular value L, this means the partial sums

S1, S2, S3, ...

get closer and closer to L. In this case, it makes sense to say that when you add those infinitely many numbers together, you get L.

If the sum doesn't get closer and closer to any particular number as you keep adding and adding, there's no reasonable way to say what the sum of those infinitely many numbers is. The series is an untamable detergent beast.

Since we promised food, we can put this in terms of a brownie, or a brownie and limits. Specifically, we're talking about the limit of the sequence of partial sums.

Start with a series and look at the associated sequence of partial sums

S1, S2, S3, ....

As with any sequence, this sequence may converge or diverge.

If the sequence of partial sums converges to a finite number L, then we say the series converges to L. In symbols, the series converges to L if there's a finite number L with

In this case we say that L is the sum of the series.

If the sequence of partial sums doesn't converge, then the series doesn't converge either, and we say the series diverges.

In terms of a brownie, our brownie continues to grow large that it won't fit inside any box, then the brownie is infinite. But if there is a box it will fit inside of, the brownie is finite in size.

Be Careful: From here on, we're using the word converge to mean three or four different things.

When talking about sequences, we use the word "converge" to mean the terms of the sequencea1, a2, a3, ...approach a limit.

When talking about series, we use the word "converge" to mean the terms of the sequence of partial sumsS1, S2, S3, ...approach a limit.

Finding the sum of a series can be difficult, because not every series has a nice formula for the partial sum Sn. Fortunately, we often only care if a series converges or diverges. That is much easier than finding the exact sum. We will see later that we have a number of tools in a handy, leather tool belt to help us figure these things out.

The finite series

3 + 4

can be rewritten as the infinite series

3 + 4 + 0 + 0 + 0 + 0 + ...

The partial sums of this series are

S1 = 3

S2 = 7

S3 = 7

and Sn = 7 for all larger n. The sequence of partial sums is

3, 7, 7, 7, 7, 7,...

Since the sequence of partial sums converges to 7, it makes sense to say that the series

Example 1

from Zeno's Paradox diverges or converges. If it converges, what is the sum of the series?

To determine if the series diverges or converges we need to look at its sequence of partial sums.

We'll start by calculating a few partial sums so we can see what they're doing:

Let's figure out the pattern so we can find the general term Sn of the sequence of partial sums.

The nth partial sum is given by

To determine if the sequence

S1, S2, S3, ...

converges, we need to know what happens to Sn as n approaches ∞. In symbols, does

exist, and if so, what is it? Since we have a formula for Sn we can answer these questions. We know

As n approaches infinity the term '-1' in the numerator becomes so small as to be irrelevant, so we're basically looking at the fraction

This means

Since the sequence of partial sums converges, the original series converges. Since the sequence of partial sums converges to 1, we say the sum of the series is 1:

Zeno's Paradox is resolved. Although there are infinitely many fractional distances between us and the brownie, all those little distances add up to 1. The math agrees with what we already know: we can indeed get to the brownie. Hopefully, it's an infinite brownie so we never have to cross this room again.

Example 2

To get from one partial sum Sn to the next, we have to step up more than 1. This means S2 must be at least 2, S3 must be at least 3, and so on.

Then

must be greater than 1, 2, 3,... and every other natural number n. This is impossible. In other words, the limit doesn't exist because the values Sn zoom off to infinity. Since the limit of the partial sums doesn't exist, the original series diverges.

This means if a ≠ 0 the series diverges. The only way the series can converge is if a = 0.

We mentioned earlier that any finite series can be thought of as an infinite series by sticking on infinitely many copies of 0 at the end.

Suppose we start cleaning out the guts of a giant pumpkin so we can carve a hideous face into it. Every day we record the hours we spend cleaning it out. It's so big that it looks like we could never finish it, but, lo and behold, one day it's cleaned out. Every day after that the number of hours we spend pumpkin prepping is 0, and our total time gourd gutting stays the same.

If we do the same for any finite series, we get some finite number L. After this point, all further partial sums will also be equal to L. Since the partial sums converge to L, the original series converges to L. Any finite series converges. We prefer our pumpkin proof over our more monotonous demonstration.